Scalar field

Homework Statement

This is not really a problem but I was going over my lecture notes and I see $\mathscr{H}=\frac{1}{2}\left(\pi^{2} + \vec{\nabla}\phi \cdot \vec{\nabla}\phi + m^{2}\phi^{2}\right)$ and $\frac{\partial\mathscr{H}}{\partial\phi} = -\nabla^{2}\phi + m^{2}\phi$

The Attempt at a Solution

I would think that $\frac{\partial\mathscr{H}}{\partial\phi} = \nabla^{2}\phi + m^{2}\phi$. But I don't know where the minus sign is coming from.

I just found a vector identity $\vec{\nabla}\phi\cdot\vec{\nabla}\phi = \vec{\nabla}\cdot\left(\phi\vec{\nabla}\phi\right) - \phi\vec{\nabla}^{2}\phi$. I now see how the result follow.

EDIT: I'm confused again. will the phi derivative of the first term vanish?

If by "the first term" you mean the $\pi^2$ term, then yes, the $\phi$ derivative will kill that term; reason being $\phi$ doesn't appear in that term, only its time derivative.

Edit: Sorry, I understand what you meant by "first term" now. The first term vanishes at infinity so you can ignore it.

Last edited:
dextercioby
If by "the first term" you mean the $\pi^2$ term, then yes, the $\phi$ derivative will kill that term; reason being $\phi$ doesn't appear in that term, only its time derivative.